The entropy conjecture for exactly Devaney chaotic maps on totally regular continua

Let $X$ be a totally regular continuum, meaning a continuum in which every two points can be joined by an arc of finite length. Write $I^{\operatorname{ED}}(X)$ for the infimum of the positive topological entropies of exactly Devaney chaotic maps on $X$. A conjecture on exactly Devaney chaotic maps.

$$I^{\operatorname{ED}}(X)=0$$

for every totally regular continuum $X$ which is not a tree. The conjecture would extend the result established in the paper for many non-tree totally regular continua, including spaces such as the Hawaiian earrings; the general assertion remains open.